Lyapunov stability of ground states of nonlinear dispersive evolution equations. (English) Zbl 0594.35005

Weinstein, Michael I.

Commun. Pure Appl. Math. 39, 51-67 (1986).

The orbital stability of the ground state solutions of the nonlinear Schrödinger equation \[ (1)\quad i\phi_ t(x,t)+\Delta \phi (x,t)+f(| \phi (x,t)|^ 2)\phi (x,t)=0,\quad x\in {\mathbb{R}}^ N \] and the generalized Korteweg-de Vries equation \[ (2)\quad w_ t(x,t)+a(w(x,t))w_ x(x,t)+w_{xxx}(x,t)=0,\quad x\in {\mathbb{R}} \] are studied using a Lyapunov function which is a conserved energy integral. In the case of (1) the ground state is orbitally stable, in the particular case where \(f(| \phi |^ 2)=| \phi |^{2\sigma}\), if \(\sigma <2/N\) and \(N=1\) or \(N=3\). A more general class of functions f is also considered. In the case of (2), if R(E) is a ground state with energy E, then a solitary wave \(\psi(x-ct)\) is orbitally stable if \(\phi(c)=(d/dc)\| R(c)\|^ 2>0.\)

Reviewer: S.P.Banks

MSC:

35B35	Stability in context of PDEs
35Q99	Partial differential equations of mathematical physics and other areas of application
35J10	Schrödinger operator, Schrödinger equation
58J47	Propagation of singularities; initial value problems on manifolds

Keywords:

Lyapunov stability; nonlinear evolution equations; solitary waves; orbital stability; ground state solutions; nonlinear Schrödinger equation; generalized Korteweg-de Vries equation; conserved energy integral

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